A family of coordinates $psi_h$ for the Teichmuller space of a compact surface with boundary was introduced in cite{l2}. In the work cite{m1}, Mondello showed that the coordinate $psi_0$ can be used to produce a natural cell decomposition of the Teichmuller space invariant under the action of the mapping class group. In this paper, we show that the similar result also works for all other coordinate $psi_h$ for any $h geq 0$.
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